Laske
a) \((-2)^3\)
b) \((-4)^2\)
c) \(-4^2\)
d) \(\sqrt{36}\)
e) \(\sqrt[4]{16}\)
f) \(-2 \cdot (-4)^2 -(-2)^3\)
Merkitse vihkoosi värikynällä pisteesi ja puuttuvat välivaiheet.
\(\require{color} \begin{align*} \textbf{a)} \quad(-2)^3 =\underbrace{(-2)\cdot (-2) \cdot (-2)}_{3 \text{kpl}} =\underline{\underline{\ -8\ }} \quad \color{red}{\text{(+1p)}} \end{align*}\)
\(\require{color} \begin{align*} \textbf{b)} \quad &(-4)^2 \\ =& \underbrace{(-4) \cdot (-4)}_{\text{ 2 kpl}}\\ =&\underline{\underline{16}} \qquad\qquad\qquad \color{red}{\text{(+1p)}} \end{align*}\)
\(\require{color} \begin{align*} \textbf{c)} \quad &-4^2 =-\underbrace{4 \cdot 4}_{\text{2 kpl}}\\ =&\underline{\underline{-16}} \qquad\qquad\qquad \color{red}{\text{(+1p)}} \end{align*}\)
\(\require{color} \begin{align*} \textbf{d)} &\sqrt{36}=\underline{\underline{\ 6\ }} \text{ (koska } 6^2=36) \quad \color{red}{\text{(+1p)}} \end{align*}\)
\(\require{color} \begin{align*} \textbf{e) } \sqrt[4]{16}=\underline{\underline{\ 2\ }} \text{ (koska } 2^4=16) \quad \color{red}{\text{(+1p)}} \end{align*}\)
\(\require{color} \begin{align*} \textbf{f)} \quad&-2 \cdot (-4)^2 -(-2)^3 \\ =&-2 \cdot (-4) \cdot (-4) - (-2) \cdot (-2) \cdot (-2) \\ =&-32+8=\underline{\underline{-24}} \qquad \color{red}{\text{(+1p)}} \end{align*}\)
Sievennä
a) \(a^2 \cdot a^6\)
b) \(\dfrac{a^2 \cdot a^7}{a^5}\)
c) \((b^3)^3\)
d) \(-2 \sqrt{3}+6 \sqrt{3}-9 \sqrt{3}\)
e) \(-5 \sqrt{5}+8 \sqrt{6}- \Big(-3 \sqrt{5}+2 \sqrt{6} \Big)\)
f) \(2 \sqrt{9}-2 \sqrt[3]{-27}+\sqrt{49}\)
Merkitse vihkoosi värikynällä pisteesi ja puuttuvat välivaiheet.
\(\require{cancel}\require{color} \begin{align*} \textbf{b)} \quad &\dfrac{a^2 \cdot a^7}{a^5} \\ =& \dfrac{\overbrace{\cancel{a}\cdot \cancel{a}}^{\text{2 kpl}} \cdot \overbrace{\cancel{a} \cdot \cancel{a} \cdot \cancel{a} \cdot a \cdot a \cdot a \cdot a}^{\text{7 kpl}}}{\underbrace{\cancel{a} \cdot \cancel{a} \cdot \cancel{a} \cdot \cancel{a} \cdot \cancel{a}}_{\text{5 kpl}} } \\ =&a^{9-5} =\underline{\underline{a^4}} \qquad \qquad\quad\color{red}\text{(+1p)} \end{align*}\)
\(\require{color} \begin{align*} \textbf{c)} \quad &(b^3)^3 \\ =& \underbrace{b^3 \cdot b^3 \cdot b^3}_{\text{ 3 kpl}} \\ =& \underbrace{\underbrace{b \cdot b \cdot b}_{\text{3 kpl}} \cdot \underbrace{b \cdot b \cdot b}_{\text{3 kpl}} \cdot \underbrace{b \cdot b \cdot b}_{\text{3 kpl}}}_{3 \cdot 3\text{ kpl}} \\ =&b^{3 \cdot 3} =\underline{\underline{b^9}} \qquad \color{red}\text{(+1p)} \end{align*}\)
\(\require{color} \begin{align*} \textbf{d)} \quad & -2 \sqrt{3}+6 \sqrt{3}-9 \sqrt{3} \\ =&\underline{\underline{-5 \sqrt{3}}} \qquad\qquad \color{red} \text{(+1p)} \end{align*}\)
\(\require{color} \begin{align*} \textbf{e)}\quad & -5 \sqrt{5}+8 \sqrt{6}- \left(-3 \sqrt{5} +2 \sqrt{6} \right) \\ =&-5 \sqrt{5}+8 \sqrt{6}+3 \sqrt{5}-2 \sqrt{6} \\ =&\underline{\underline{-2 \sqrt{5}+6 \sqrt{6}}} \qquad \quad\color{red} \text{(+1p)} \end{align*}\)
\(\require{color} \begin{align*} \textbf{f) }\quad &2 \sqrt{9}-2 \sqrt[3]{-27}+\sqrt{49} \\ =&2 \cdot 3 - 2 \cdot (-3) + 7 \\ =&6+6+7 \\ =&\underline{\underline{\ 19\ }} \qquad\qquad\qquad\color{red} \text{(+1p)} \end{align*}\)
Laske
a) \(5-2^3\)
b) \((5-2)^3\)
c) \(\dfrac{12^3}{4^3}\)
d) \(0,4^4 \cdot 5^4\)
e) \(\sqrt{2} \cdot \sqrt{32}\)
f) \(\dfrac{\sqrt[3]{54}}{\sqrt[3]{2}}\)
Merkitse vihkoosi värikynällä pisteesi ja puuttuvat välivaiheet.
\(\require{color} \begin{align*} \textbf{a)}\quad &5-2^3\\ =&5-\underbrace{2 \cdot 2 \cdot 2}_{\text{3 kpl}} \\ =&5-8 \\ =&\underline{\underline{-3}} && \color{red}\text{(+1p)} \end{align*}\)
\(\require{color} \begin{align*} \textbf{b) }\quad & (5-2)^3 \\ =&3^3 \\ =&3 \cdot 3 \cdot 3 \\ =&\underline{\underline{\ 27\ }} &&\color{red}\text{(+1p)} \end{align*}\)
\(\require{color} \begin{align*} \textbf{c)}\quad & \dfrac{12^3}{4^3} && \Big|\Big| \Big(\dfrac{a}{b}\Big)^n=\dfrac{a^n}{b^n}\\ =&\Big(\dfrac{12}{4} \Big)^3 \\ =&3^3 \\ =&3 \cdot 3 \cdot 3 \\ =&\underline{\underline{\ 27\ }} && \color{red}\text{(+1p)} \end{align*}\)
\(\require{color} \begin{align*} \textbf{d)} \quad &0,4^4 \cdot 5^4 && \Big| \Big| (a \cdot b)^n = a^n \cdot b^n \\ =&(0,4 \cdot 5)^4 \\ =&2^4 \\ =&2 \cdot 2 \cdot 2 \cdot 2 \\ =&\underline{\underline{16}} && \color{red}\text{(+1p)} \end{align*}\)
\(\require{color} \begin{align*} \textbf{e)} \quad &\sqrt{2} \cdot \sqrt{32} \qquad \quad | | \sqrt[n]{a} \cdot \sqrt[n]{b}=\sqrt[n]{a \cdot b} \\ =&\sqrt{2 \cdot 32} \\ =&\sqrt{64} \\ =&\underline{\underline{\ 8\ }} \quad \text{ (koska } 8^2=64) \qquad \color{red}\text{(+1p)} \end{align*}\)
\(\require{color} \begin{align*} \textbf{f)} \quad &\dfrac{\sqrt[3]{54}}{\sqrt[3]{2}} \qquad\qquad \Big| \Big| \dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\dfrac{a}{b}} \\ =&\sqrt[3]{\dfrac{54}{2}} \\ =&\sqrt[3]{27} \\ =&\underline{\underline{\ 3\ }} \text{ (koska } 3^3=27) \qquad \color{red}\text{(+1p)} \end{align*}\)
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